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A215363 E.g.f. satisfies: A(x) = x + A(x)^2*cosh(A(x)). 1
1, 2, 12, 132, 2040, 40350, 974400, 27805736, 915505920, 34160797530, 1424581678080, 65660547312492, 3314551571595264, 181866769617012662, 10777121944589844480, 685937077729538151120, 46668919680893409361920, 3380042082757952844150066, 259638732115410022642483200 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..19.

FORMULA

E.g.f. satisfies:

(1) Series_Reversion(x - x^2*cosh(x)).

(2) x + Sum_{n>=1} d^(n-1)/dx^(n-1) cosh(x)^n*x^(2*n) / n!.

(3) x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) cosh(x)^n*x^(2*n-1) / n! ).

a(n) = [x^n/n!] x/(1 - x*cosh(x))^n / n for n>0.

a(n) ~ s*sqrt(r/(6*r-2*s-r*s^2+s^3)) * n^(n-1) / (exp(n) * r^n), where s = 0.4227473416936597149... is the root of the equation s*(2*cosh(s) + s*sinh(s)) = 1, and r = s - s^2*cosh(s) = 0.22782318947143997934... - Vaclav Kotesovec, Jan 13 2014

EXAMPLE

E.g.f: A(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2040*x^5/5! +...

Series expressions:

A(x) = x + cosh(x)*x^2 + d/dx cosh(x)^2*x^4/2! + d^2/dx^2 cosh(x)^3*x^6/3! + d^3/dx^3 cosh(x)^4*x^8/4! +...

log(A(x)/x) = cosh(x)*x + d/dx cosh(x)^2*x^3/2! + d^2/dx^2 cosh(x)^3*x^5/3! + d^3/dx^3 cosh(x)^4*x^7/4! +...

MATHEMATICA

Rest[CoefficientList[InverseSeries[Series[x - x^2*Cosh[x], {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 13 2014 *)

PROG

(PARI) {a(n)=n!*polcoeff(serreverse(x-x^2*cosh(x+x*O(x^n))), n)}

for(n=1, 25, print1(a(n), ", "))

(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, cosh(x+x*O(x^n))^m*x^(2*m)/m!)); n!*polcoeff(A, n)}

(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, cosh(x+x*O(x^n))^m*x^(2*m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}

(PARI) {a(n)=n!*polcoeff(x/(1 - x*cosh(x+x*O(x^n)))^n/n, n)}

CROSSREFS

Cf. A215364, A213643.

Sequence in context: A258467 A286422 A073551 * A200319 A213640 A266489

Adjacent sequences:  A215360 A215361 A215362 * A215364 A215365 A215366

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 08 2012

STATUS

approved

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Last modified October 23 00:39 EDT 2021. Contains 348211 sequences. (Running on oeis4.)